Kinematics

Kinematics describes motion with positions, angles and velocities only. It ignores mass and force. Its partner, dynamics, adds them: how torques and forces make the body accelerate.

A pose that is kinematically solved is geometrically right (the feet are where they should be, the joints within range) but nobody has checked that the body could actually hold it under gravity. Making it physically work is a separate job, here done by the policy.

Forward kinematics

Forward kinematics answers: given every joint angle $q$, where is each body part? It’s a chain of rotations and offsets from the root outward. For a two-segment planar arm with lengths $l_1, l_2$ and joint angles $q_1, q_2$, the hand is at

$$\begin{aligned} x &= l_1 \cos q_1 + l_2 \cos(q_1 + q_2) \\ y &= l_1 \sin q_1 + l_2 \sin(q_1 + q_2) \end{aligned}$$

In 3D each joint applies a rotation, and the position of a body part is the product of every transform from the pelvis out to it:

$$T_{\text{hand}} = T_{\text{pelvis}}\; T_1(q_1)\; T_2(q_2) \cdots T_n(q_n)$$

Forward kinematics always has exactly one answer and is cheap to compute. Going the other way is harder: see inverse kinematics.

In this project

The stance and rhythm are first solved kinematically: the joint angles that put the feet, hands and centre of mass where the coach wants them. That kinematic reference motion is what the policy then learns to track with real physics.